B737 pilot reported the autopilot and flight control system malfunctioned several times during arrival descent resulting in periods of loss of aircraft control and altitude deviation. Flight crew continued descent and systems appeared to stop malfunctioning.

Narrative

While descending into ZZZ on flight XXXX. We experienced a flight control or automation anomaly. While flying in LNAV; VNAV and initially having a VNAV PATH. our autopilot and flight control system erroneously started flying to 340 knots indicated and pitching as such to speed the aircraft up. Although only momentarily we saw sink rate values between 6;000 and 8;000 FPM. While trying to establish what was occurring. The First Officer (correctly in my mind disconnected the autopilot and hand flew the aircraft.) When we thought we had control of aircraft we elected to try the autopilot again. However; it reverted back to its ways and again began shooting for a target airspeed of 338 KTS. and began descending rapidly. Once again after disconnecting the auto pilot we could then regain control of the MCP (Mode Control Panel). This time after 'dumying down' the control modes; we were able to get the aircraft to respond to our inputs and Level change commands began. After this point we were able to re-establish a normal descent profile and speed targets. So decided to continue to ZZZ and attempt a landing at that time. Speed targets and crossing restrictions from this point on were met and the aircraft remained controllable throughout the landing. Sadly; while this event was occurring; we missed the target altitudes on the ZZZZZ arrival and crossed 'ZZZZZ' way point significantly low. At around 15;800 instead of a bottom crossing altitude of 17000. Although this was a significant control startle factor. The aircraft never generated a master caution or warning message. At the very onset of the event; I the Captain was making a PA; informing the passengers that due to the higher winds I was expecting some stronger turbulence and to also request that the flight attendants prepared the cabin for landing and return to their seats for their safety. The ATC controller who was working our flight initially asked what altitude we were at; and said I show you low for the crossing altitude; however then came back on and was asking if we were alright; as he could apparently see our altitude changes. I initially responded to him saying we had a flight control or automation issue occurring. But then just prior to being handed off to ZZZ Approach informed him we had control of the ATC and all seemed to be returning to normal.I cannot attest to it factually or not; but we were wondering if the extreme winds both in the Jetstream and low altitude regimes of flight played a role in the incident. Even when I go to make a PA. I cannot allow the attention to drift away from the flight to the other tasks I am to complete either as the PF or PM